We learn that correct
thinking by logic is a reduction by syllogism. All men are mortal. Socrates is a man. Therefore, Socrates
is mortal. If A =B and B = C, then
A = C. The middle or the mean term is factored out. Boole granted that syllogism is one kind of valid method. However, rather than reducing
statements, Boole argued by expansion for a deeper and extended understanding
of seemingly simple truths. His
algebra is deceptively similar to – but different from – the forms we learned
in high school; and so the learner here must pay close attention to these laws
of thought.
If x is “red” then xx
means “red red” which is nonetheless still “red.” If the essential attribute under consideration is “red” then
red red things with their redness removed do not exist: xx – x = 0. Taking from Leibniz, Boole lets 0 be non-existence and 1 be the universe. If x is our subject, then (1-x) is everything else in the universe except x. Boole rewrites xx-x as x(1-x) = 0 to give Aristotle’s Law of Contradiction: A thing cannot
both have and not have the same attribute in the same way at the same
time. Boole calls this “duality.” Also following Leibniz, Ayn Rand delivered that as “A or non-A.”
In 400 pages, Boole
carefully applies the essential truths he discovered. He extracts more meaning –
and more exacting meanings – from common statements. However, his insistence on closely laying every step of this
long journey, couched, as it was, and, indeed, of necessity, both historical
and personal, must have been, in Victorian prose, renders this important work so
Germanic as to be nearly unreadable.
It is well worth the effort.
“Wealth consists of things
transferable, limited in supply, and either productive of pleasure or
preventive of pain.”
w = wealth
s = limited in supply
t = transferable
p = productive of pleasure
r = preventive of pain
Boole specially defines
0/0 to mean the undefined. He
includes this to stand for the indefinite
class: those things not included in the present statement but which exist
nonetheless.
Given that this book is
400 pages and Ayn Rand’s Introduction to
the Objectivist Epistemology is 164, Boole boldly goes beyond the orbit of
concept formation. He nonetheless
is in the same material space: he identifies reality; and ties concepts to
objects. Boole explores the relationships between the processes of thought and
perception and the expression of them through language.
Boole even applies his
method to statements about God.
His goal is not to address God’s existence but to expand and analyze statements
about God asserted by Baruch (“Benedict”) Spinoza and Samuel Clarke. (“… considered the major figure in
British philosophy between John Locke and George Berkeley.” – Wikipedia
here.) From there, he examines
Aristotle’s logic and Fermat’s probabilities. The last third
of the book is about the logic of probability theory.
ALSO ON NECESSARY FACTS